On Functors Expressible in the Polymorphic Typed Lambda
Calculus

J.C. Reynolds and G.D. Plotkin

Abstract: Given a model of the polymorphic typed lambda
calculus based upon a Cartesian closed category K, there
will be functors from K to K whose action on objects
can be expressed by type expressions and whose action on morphisms
can be expressed by ordinary expressions. We show that if T
is such a functor then there is a weak initial T-algebra and
if, in addition, K possesses equalizers of all subsets of
its morphism sets, then there is an initial T-algebra. It
follows that there is no model of the polymorphic typed lambda
calculus in which types denote sets and S -> S'
denotes the set of all functions from S to S'.